Find the x-value(s) of the relative maxima and relative minima

User Generated

ZvpunrynBP

Mathematics

Description

User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

Thank you for the opportunity to help you with your question!

First, we rewrite the 32/x by using this power law:   1/x = 1/x^1 = x^-1

f(x) = 2x^2 + 32/x + 7  ------------> f(x) = 2x^2 + 32x^-1 + 7

Then we find the derivative of the function. Let's remember the derivative of a power:   (x^n)' = n*x^n-1 ; where n is the exponent of the x. By the way, let's remember also that the derivative of any constant is 0.

f'(x) = (2x^2 + 32x^-1 + 7)'  ------> f'(x) = (2x^2)' + (32x^-1)' + (7)' ----> f'(x) = 2(x^2)' + 32(x^-1)' + (7)'

f'(x) = 2(2x^2-1) + 32(-1x^-1-1) + 0  --------> f'(x) = 4x^1 - 32x^-2   --------> f'(x) = 4x - 32x^-2

f'(x)= 4x - 32/x^2

Then we equal the derivative to zero and solve for x.

4x - 32/x^2 = 0  -----------> 4x*x^2 - 32/x^2 * x^2 = 0 ----------->  4x^3 - 32 = 0 ------------> 4x^3 - 32 + 32 = 0 + 32

4x^3 = 32  ----------->  4x^3/4 = 32/4 ------------> x^3 = 8   ------------>  (x^3)^1/3 = (8)^1/3  --------> x = 8^1/3

Or x = cubic root(8) ---------->  x = 2

Then we find the second derivative.

f''(x) = (4x - 32x^-2)' = (4x)' - (32x^-2)' = 4(x)' - 32(x^-2)' = 4(1) - 32(-2)x^-2-1 = 4 + 64x^-3

f''(x) = 4 + 64/x^3

Then we enter x = 2 into the second derivative and if it is greater than zero (positive) then it is a minimum

and if it is less than zero (negative) then it is a maximum.

f''(2) = 4 + 64/2^3 = 4 + 64/8 = 4 + 8 = 12 > 0 (positive). Then x = 2 is a minimum and we don't have any maximum.

Relative maxima: DNE (we don't have it).

Relative minima: x = 2

Please let me know if you have any doubt or question.

Please let me know if you need any clarification. I'm always happy to answer your questions.


Anonymous
This is great! Exactly what I wanted.

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags